Solving Cauchy Issues of Highly Nonlinear Elliptic Equations Using a Meshless Method

In this paper, we address 3D inverse Cauchy issues of highly nonlinear elliptic equations in large cuboids by utilizing the new 3D homogenization functions of different orders to adapt all the specified boundary data. We also add the average classification as an approximate solution to the nonlinear operator part, without requiring to cope with nonlinear equations to resolve the weighting coefficients because these constructions are owned many conditions about the true solution. The unknown boundary conditions and the result can be retrieved straightway by coping with a small-scale linear system when the outcome is described by a new 3D homogenization function, which is right to find the numerical solutions with the errors smaller than the level of noise being put on the over-specified Neumann conditions on the bottom of the cuboid. Besides, note that the new homogenization functions method (HFM) does not require dealing with the regularization and highly nonlinear equations. The robustness and accuracy of the HFM are verified by comparing the recovered results of several numerical experiments to the exact solutions in the entire region, even though a very large level of noise 50% is imposed on the over specified Neumann conditions. The numerical errors of our scheme are in the order of O(10−1)–O(10−4).